So basically he said that we should never add each other on facebook. When I asked him why he was like 'well I can get in touch with you through email and texts, or call you and see you, why use facebook then'

I was like okay, well, it's fine, but it bothers me and I don't understand why he doesn't want to.
Yes, it's just facebook but he goes around saying that I'm his girlfriend and it's weird to have to explain it to people. Plus, I'm worried that he might be hiding something away from me.

Seems strange. If you have facebook, you use it to have your friends on it, and nobody is closer as a friend than your gf. If he didn't want you listed as his gf, rather have "in a relationship" only - fine - but not even have you added? Doesn't sound good.